Nuclear magnetic resonance is observed in materials containing nuclei with non-zero spin angular momentum. Associated with their spin angular momentum, S, these nuclei possess a magnetic dipole moment, μ. In a static magnetic field, B0, these dipole moments have a discrete set of spin eigenstates which are populated according to the Boltzmann distribution. This results in a net magnetisation, M, parallel to B0 which can be manipulated by radiofrequency (RF) pulses at the Larmour frequency. If the net magnetisation is rotated away from the direction of B0 by an RF pulse, there is a non-zero transverse magnetisation that precesses at the Larmour frequency, ω, resulting in a signal that can be detected with an RF coil. This signal decays away over time due to both spin-lattice and spin-spin relaxation mechanisms.
To form an image of the object the signal can be localised in space by applying linear magnetic field gradients. These result in known linear variations in the static magnetic field as a function of position. A series of measurements of the signal are performed during which the gradients are altered according the particular localisation method being employed. The image is then calculated from these measurements using one of several well known reconstruction techniques.
Slice- (or slab-) selection is commonly used in MRI and MRS to excite the magnetisation over a volume (“slice”) of thickness Δz. A slice-selection gradient is applied, which causes the precession frequency of the net magnetisation to vary linearly with spatial position. An RF pulse with bandwidth, Δf, is applied concurrently with the slice-selection gradient, at an amplitude, Gz, such that only the net magnetisation within the slice is rotated away from the direction of B0. The thickness of the slice, Δz, is related to the bandwidth, the amplitude of the slice-selection gradient, Gz, and the gyromagnetic ratio, γ, by:
                              Δ          ⁢                                          ⁢          z                =                              2            ⁢            πΔ            ⁢                                                  ⁢            f                                γ            ⁢                                                  ⁢                          G              z                                                          (        1        )            
If the static magnetic field is spatially inhomogeneous, the resulting data can suffer from signal dropout. Signal dropout is a particular problem for data acquired with gradient-echo and asymmetric spin-echo techniques. Localised inhomogeneities may result from magnetic susceptibility gradients at the boundaries within the object being imaged. These localised magnetic field inhomogeneities occur for example near bone, soft-tissue and air interfaces in the human head causing signal dropout in the images of the brain in the vicinity of these areas. The signal dropout is caused by dephasing of the transverse magnetisation by the susceptibility gradients.
A number of techniques have been suggested to reduce the problem of signal dropout caused by susceptibility gradients. These include reducing the echo time or the voxel volume, localised magnetic field shimming, dynamic shim updating, the use of passive shims constructed from diamagnetic materials, localised active shimming, z-shimming, and compensation gradients in the frequency and phase encoding directions.
An alternative approach is to use tailored RF pulses for slice selection. Z. H. Cho and Y. M. Rho, “Reduction of Susceptibility Artifact in Gradient-Echo Imaging”, Magnetic Resonance in Medicine, 23, 193-200, 1992 [1], introduced the concept that the phase dispersion caused by a linear susceptibility gradient could be partially cancelled using a tailored RF pulse, therefore reducing signal dropout. The tailored RF pulse induces a quadratic (or approximately quadratic) phase variation in the transverse magnetisation in the direction of slice selection, φRF(Z) as given by equation 2.φRF(z)=az2  (2)
Here, the parameter a governs the degree of quadratic variation in the phase of the transverse magnetisation induced by the RF pulse. In addition Cho and Rho developed an analytical model describing how the MR signal varies as a function of the susceptibility gradient and the parameter a. The model was used to determine the value of the parameter a needed to reduce signal dropout. In addition, it highlighted that there is a trade-off between signal recovery in areas with susceptibility gradients and a reduction of signal in areas of homogeneous magnetic field when RF pulses which induce a quadratic phase variation in the phase of the transverse magnetisation are used. Cho and Rho demonstrated that the problem of signal loss in images of the human head could be reduced using such a pulse. However they did not specify how the RF pulse could be produced in practice; neither an algorithm nor functional form was described. In addition, the analytical model they described was based on the simplifying assumption that the excited slices in MRI have perfectly rectangular slice profiles. This is not achievable in practice and as such their model is inaccurate and therefore cannot be used to accurately determine the degree of quadratic phase variation, a, that the RF pulse needs to induce to reduce signal dropout from a particular range of susceptibility gradients. Three further manifestations of tailored RF pulses, all based on the concept described by Cho et al., have been described.
J. Y. Chung, H. W. Yoon, Y. B. Kim, H. W. Park, Z. H. Cho, “Susceptibility Compensated fMRI Study Using a Tailored RF Echo Planar Imaging Sequence”, Journal of Magnetic Resonance Imaging, 29, 221-228, 2009 [2], used the analytical model of Cho et al. to select the degree of quadratic phase variation, a, induced by the RF pulse such that the signal loss in areas of homogeneous magnetic field was reduced relative to the previously published implementation, although at a cost of less effective recovery of signal dropout. It should be noted however that they found large discrepancies between the signal recovery found experimentally and that predicted theoretically by the model of Cho et al, reinforcing the claim made above that the model of Cho et al. is inaccurate and therefore should not be used to determine the optimal degree of quadratic phase variation, a. Again, Chung et al. did not specify how the RF pulse they used could be produced in practice.
Both J. Mao and A. W. Song, “Intravoxel rephasing of spins dephased by susceptibility effect for EPI sequences”, In: Proceedings of the International Society for Magnetic Resonance in Medicine, 1999, Philadelphia. p. 1982[3], and K. Shmueli, D. L. Thomas and R. Ordidge, “Signal Drop-Out Reduction in Gradient Echo Imaging with a Hyperbolic Secant Excitation Pulse—An Evaluation Using an Anthropomorphic Head Phantom”, In: Proceedings of the International Society for Magnetic Resonance in Medicine, 2006 Seattle. p. 2385[4], used complex hyperbolic secant RF pulses to generate approximately quadratic phase profiles in the transverse magnetisation. Such pulses, with duration TRF, have a magnetic field B1 perpendicular to the static magnetic field B0 that varies with time, t, according to:B1(t)=[A0 sec h(βt)]1+iμ  (3)
In equation 3, A0 is the peak amplitude of the RF pulse, β is known as the modulation angular frequency,
            -                        T          RF                2              <    t    <                  T        RF            2        ,      i    =                  -        1            and as shown by Shmueli et al. the parameter μ is proportional to the degree of quadratic phase variation, a. Additionally Mao et al. tailored the area of the slice-refocusing gradient (a procedure referred to as z-shimming, as first suggested by J. Frahm, M. Klaus-Dietmar, and H. Wolfgang, “Direct FLASH MR imaging of magnetic field inhomogeneities by gradient compensation”, Magnetic Resonance in Medicine, 6, 4, 474-480, 1988 [5]) in an effort to further reduce signal dropout. Neither Shmueli et al. nor Mao et al. described a method to systematically determine the parameters of the RF pulse or the accompanying slice-selection and slice-refocusing gradients to correct the problem of signal dropout caused by a specific range of linear susceptibility gradients. In addition, both Shmueli et al. and Mao et al. incorrectly assumed that the frequency bandwidth of the pulse (used to determine the slice-selection gradient amplitude for a given slice thickness) was given by
            μ      ⁢                          ⁢      β        π    .